This document discusses the generalization of the fractional Fourier transform (FrFT). It begins by introducing the conventional FrFT and defining the necessary test function space. It then proves three key properties for the generalized FrFT: 1) the scaling property, which shows how the transform is affected by scaling the input, 2) Parseval's identity, which establishes the energy conservation of the transform, and 3) the modulation property, which describes how modulating the input affects the transformed output. The document concludes by stating that modulation theorem, scaling property, and Parseval's theorem have been proved for the generalized FrFT.

1. V.D.Sharma Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.1463-1467
RESEARCH ARTICLE
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Modulation and Parsvels Theorem For Generalized Fractional
Fourier Transform
V. D. Sharma
Mathematics Department, Arts, Commerce and Science College, Amravati, 444606 (M.S.) India
Abstract
The fractional Fourier transform (FrFT) was first introduced as a way to solve certain classes of ordinary and
partial differential equations arising in quantum mechanics. FrFt has established itself as a powerful tool for the
analysis of time-varying signals, especially in optics. FrFt has found applications in areas of signal processing
such as repeated filtering, fractional convolution and correlation, beam forming, optimal filter, convolution,
filtering and wavelet transforms, time frequency representations.
In this paper operational calculus for
Generalized fractional Fourier transform are obtained.
Keywords: Generalized function, Fractional Fourier transform, Fourier transform, signal processing, Test
function space.
I.
Introduction
The Fourier transform is one of the most
important mathematical tools used in Physical optics,
linear system theory, signal processing and so on [1].
The conventional Fourier transform can be regarded as
a rotation in the time-frequency plane and the FrFT
performs a rotation of signal to any angle. Moreover,
fractional Fourier transform serves as an orthonormal
signal representation for chirp signal. The Fractional
Fourier transform is also called rotational Fourier
transform or angular Fourier transform in some
documents. The FrFT is also related to other timevarying signal processing tools such as Wigner
distribution, Short time Fourier transform, Wavelet
transform [2].
In our previews work we have defined in [3, 4] as,
1.1. Conventional fractional Fourier transform:
The FrFT with parameter
of
denoted
by
performs a linear operation given by
Where
1.2 The test function space E:
An infinitely differentiable complex valued
smooth function on
belongs to
if for each
compact set
, where
0, I Rn.
, where p=1, 2, 3.....
Thus
will denote the space of all
with suppose contained in . Note that the
space E is complete and therefore a Frechet space.
Moreover we say that
is a fractional Fourier
transformable if it is a member of , the dual space of
.
FrFT has many applications in solution of differential
equations, optical beam propagtioan and spherical
mirror resonators, optical diffraction theory, quantum
mechanics, statistical optics, signal detectors, pattern
recognition, space image recovery etc. [5, 6].
In the present work Generalization of the FrFT is
presented by kernel method. Scaling property,
Modulation theorem, and Parsvels theorem is proved
for Generalized Fractional Fourier transform.
II.
Distributional Fractional Fourier
Transform
The
transforms of
Distributional
fractional
Fourier
can be defined by
where
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2. V.D.Sharma Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05
The right hand side of (3.1) has a meaning as the application of
III.
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to
.
Scaling Property
where
Proof:
putting
where
Parseval’s Identity for Fractional Fourier transform
IV.
If
and
then
i.
ii.
Proof: By definition of
using inversion formula of
Taking complex conjugate we get
Now
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3. V.D.Sharma Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05
Hence proved
ii) Let
eqn (4.2) becomes
then
If
denotes generalized fractional Fourier transform of
and
V.
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then
Modulation property for FrFT
then
Proof:
6. If
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denotes generalized fractional Fourier transform of
then
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4. V.D.Sharma Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05
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Proof:
Generalized Fractional Fourier Transform
Sr.
No
1.
2.
3.
4.
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5. V.D.Sharma Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05
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5.
6.
7.
8.
VI.
Conclusion
I
n this paper Modulation theorem, Scaling property,
Parsvels theorem is proved for Generalized Fractional
Fourier transform.
References
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[3].
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[5].
[6].
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[8].
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V.D.Sharma,
Generalization
of
Twodimensional Fractional Fourier transforms,
Asian Journal of current Engineering and
Maths, 46-48, 2012.
V.D.Sharma, operation transform Formulae on
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15-18, 2012.
Bultheel A. et.al., Recent developments in the
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Felipe Salazon, An Introduction to the
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applications, Dec. 7, (2011) 1-8
Zemanian
A.H.,
Generalized
Integral
transform, Interscience Publication, New
York, 1968.
B.N.Bhosale, Integral transformation of
Generalized functions. Discovery publishing
hour, 2005.
Ervin S., Djurovic I. et.al., fractional Fourier
transform as a signal processing tool; an
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